Global exponential stability for a class of retarded functional differential equations with applications in neural networks

In this paper, the global exponential stability and asymptotic stability of retarded functional differential equations with multiple time-varying delays are studied by employing several Lyapunov functionals. A number of sufficient conditions for these types of stability are presented. Our results show that these conditions are milder and more general than previously known criteria, and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Furthermore, the results obtained for neural networks with time-varying delays do not assume symmetry of the connection matrix.

[1]  Kwok-Wo Wong,et al.  Novel stability conditions for Cellular Neural Networks with Time Delay , 2001, Int. J. Bifurc. Chaos.

[2]  C. Leung,et al.  Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function , 2001 .

[3]  R. Westervelt,et al.  Stability of analog neural networks with delay. , 1989, Physical review. A, General physics.

[4]  H Lu,et al.  Stability criteria for delayed neural networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  M. Markus,et al.  Oscillations and turbulence induced by an activating agent in an active medium. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Zhang Yi,et al.  Global exponential stability and periodic solutions of delay Hopfield neural networks , 1996, Int. J. Syst. Sci..

[7]  Jinde Cao,et al.  Global stability analysis in delayed cellular neural networks. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Guanrong Chen,et al.  Novel robust stability criteria for interval-delayed Hopfield neural networks , 2001 .

[9]  Pauline van den Driessche,et al.  Global Attractivity in Delayed Hopfield Neural Network Models , 1998, SIAM J. Appl. Math..

[10]  J. Hopfield,et al.  Computing with neural circuits: a model. , 1986, Science.

[11]  Mark P. Joy,et al.  Results concerning the absolute stability of delayed neural networks , 2000, Neural Networks.

[12]  Xiaofeng Liao,et al.  Robust stability for interval Hopfield neural networks with time delay , 1998, IEEE Trans. Neural Networks.

[13]  Xiaofeng Liao,et al.  Novel stability criteria for bidirectional associative memory neural networks with time delays , 2002, Int. J. Circuit Theory Appl..

[14]  X. Liao,et al.  Stability analyses of cellular neural networks with continuous time delay , 2002 .

[15]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[16]  M. Joy.  On the Global Convergence of a Class of Functional Differential Equations with Applications in Neural Network Theory , 1999 .

[17]  Masahiko Morita,et al.  Associative memory with nonmonotone dynamics , 1993, Neural Networks.

[18]  X. Liao,et al.  Bifurcation analysis on a two-neuron system with distributed delays , 2001 .

[19]  Wang,et al.  Global stability and local stability of Hopfield neural networks with delays. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Rossana Vermiglio,et al.  Stability of some test equations with delay , 1994 .

[21]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Amir F. Atiya,et al.  How delays affect neural dynamics and learning , 1994, IEEE Trans. Neural Networks.

[23]  M. Forti.  On Global Asymptotic Stability of a Class of Nonlinear Systems Arising in Neural Network Theory , 1994 .

[24]  Jinde Cao,et al.  Stability analysis of delayed cellular neural networks , 1998, Neural Networks.

[25]  K. Gopalsamy,et al.  Stability in asymmetric Hopfield nets with transmission delays , 1994 .

[26]  Xue-Zhong He,et al.  Delay-independent stability in bidirectional associative memory networks , 1994, IEEE Trans. Neural Networks.

[27]  Leon O. Chua,et al.  Stability and dynamics of delay-type general and cellular neural networks , 1992 .

[28]  Juebang Yu,et al.  Qualitative analysis of Bi‐directional Associative Memory with time delay , 1998 .

[29]  Leon O. Chua,et al.  Stability of cellular neural networks with dominant nonlinear and delay-type templates. (Memo UCB/ERL No. M92/121.) , 1993 .

[30]  Shun-ichi Amari,et al.  Exponential Convergence of Delayed Dynamical Systems , 2001, Neural Computation.

[31]  Masahiko Morita,et al.  Capacity of associative memory using a nonmonotonic neuron model , 1993, Neural Networks.

[32]  S. Arik.  Stability analysis of delayed neural networks , 2000 .

[33]  Hongtao Lu,et al.  On stability of nonlinear continuous-time neural networks with delays , 2000, Neural Networks.